Problem 26 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 9-28, find the convergence set for the given power series. Hint: First find a formula for the nth term; then use the Absolute Ratio Test. $$ \frac{x-2}{1^{2}}+\frac{(x-2)^{2}}{2^{2}}+\frac{(x-2)^{3}}{3^{2}}+\frac{(x-2)^{4}}{4^{2}}+\cdots $$

Step-by-Step Solution

Answer

The convergence set of the series is \([1, 3]\).

Step 1: Identify the General Term

The given series is of the form \( \sum_{n=1}^{\infty} \frac{(x-2)^{n}}{n^{2}} \). Each term of the series can be written as \( a_n = \frac{(x-2)^{n}}{n^{2}} \).

Step 2: Apply the Ratio Test

To determine the convergence set, we employ the Ratio Test. Compute the ratio of successive terms: \( \left| \frac{a_{n+1}}{a_n} \right| = \left| \frac{(x-2)^{n+1}}{(n+1)^{2}} \cdot \frac{n^{2}}{(x-2)^{n}} \right| = \frac{|x-2|}{(1+\frac{1}{n})^{2}} \).

Step 3: Evaluate the Limit

Take the limit as \( n \to \infty \): \( \lim_{n \to \infty} \frac{|x-2|}{(1+\frac{1}{n})^2} = |x-2| \). According to the Ratio Test, the series converges if \( |x-2| < 1 \).

Step 4: Convergence Interval

The inequality \( |x-2| < 1 \) translates into \( 1 < x < 3 \). This provides the interval of x for which the series converges.

Step 5: Endpoint Testing

To find if the series converges at the endpoints, substitute \( x = 1 \) and \( x = 3 \) into the series and evaluate each separately. At \( x = 1 \), the series becomes \( \sum_{n=1}^{\infty} \frac{(-1)^{n}}{n^{2}} \), which converges by the Alternating Series Test. At \( x = 3 \), the series becomes \( \sum_{n=1}^{\infty} \frac{1}{n^{2}} \), which converges by the p-series test with \( p = 2 \).

Step 6: Final Convergence Set

Since the series converges at both ends, the convergence interval is closed: \([1, 3]\).

Key Concepts

Ratio Test

The Ratio Test is a crucial method for studying the convergence of series, particularly power series. For any sequence of terms, \(a_n\), the Ratio Test involves looking at the limit of the absolute value of the ratio of successive terms: \[\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\]This test states that if this limit is less than 1, the series converges absolutely. If the limit is greater than 1, the series diverges. Importantly, if the ratio is exactly 1, the test is inconclusive, and other methods must be employed.

• Convergence if \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| < 1 \)
• Divergence if \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| > 1 \)
• Inconclusive if \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = 1 \)

In our problem, applying the Ratio Test involves simplifying the ratio of successive terms of the power series, ultimately examining the behavior as \(n\) becomes very large.

Convergence Interval

The Convergence Interval is the range of values \(x\) for which a power series converges. This is pivotal in understanding where a series is valid. After applying the Ratio Test, if we find a condition like \(|x - 2| < 1\), it points to the fact that the series behaves nicely between two specific x-values.This condition translates neatly into the interval \(1 < x < 3\). This interval identifies all \(x\)-values that make the series converge based solely on the application of the Ratio Test. However, checking endpoints is needed for completeness, as they might sometimes be included or excluded, changing the interval to a closed or open one.

Endpoint Testing

Endpoint Testing adds depth to the Convergence Interval by determining what happens exactly at the endpoints of the interval. In this context, we need to substitute back into the original series for these specific x-values.For example:

• At \(x = 1\), the series becomes \(\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n^{2}}\). Here, the Alternating Series Test is applicable.
• At \(x = 3\), the series turns into \(\sum_{n=1}^{\infty} \frac{1}{n^{2}}\), allowing the use of the p-series Test.

Endpoint testing is a necessary step for finalizing whether the series converges exactly at the endpoints, turning an open interval into a potentially closed one depending on the results.

Alternating Series Test

The Alternating Series Test is quite handy when checking convergence of series like our example at \(x = 1\). An alternating series changes its sign with every subsequent term and is defined as \( (-1)^n a_n \), repeated for each term.For convergence, two conditions must be satisfied:

• The absolute values of the terms \(a_n\) must decrease steadily, \(a_{n+1} < a_n \) for all \(n\).
• The limit of \(a_n\) as \(n\) approaches infinity must equal zero, \(\lim_{n \to \infty} a_n = 0\).

For our series at \(x = 1\), it meets these criteria since \(\frac{1}{n^2}\) decreases and converges to zero. Thus, the series converges at this endpoint, fully validating the interval.

p-series Test

The p-series Test is a reliable tool when assessing convergence. A typical p-series is of the form \( \sum_{n=1}^{\infty} \frac{1}{n^p} \). The test focuses on the value of the exponent \(p\):

• If \(p > 1\), the series converges.
• If \(p \leq 1\), the series diverges.

In our scenario, when you check the series at \(x = 3\), it turns into \( \sum_{n=1}^{\infty} \frac{1}{n^2} \). Clearly, \(p = 2\), which is greater than 1, confirming that the series converges. This final check supports the claim that the series covers the interval from \([1, 3]\), ensuring that both endpoints are indeed points of convergence.